Somewhat more seriously, it's strongly implied in TWOK that Khan's lust for revenge is fueled by his helplessness to protect his people - including and especially his "beloved wife" - from the environmental disaster on Ceti Alpha V. He places the blame for that on Kirk. It sounds as if he cared very much.

I have to agree with you on that, after Joachim died in his arms on the Reliant, Khan did say 'I will avenge you."

Of course, there's the theory that's been bandied about that Joachim was supposed to be Khan's son, but the impression that I got was that he was a trusted friend and probably Khan's right-hand man.

According to the Star Trek Encyclopedia: Joachim: Aide to Khan Noonien Singh, and one of the 96 surviving genetic "supermen" who escaped from Earth in 1996 aboard the S.S. Botany Bay.
Joachim served as Khan second-in-comand when Khan commandeered the Starship Reliant and died when Khan detonated the Genesis Device.